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By using the truncation technology, a kind of fast incomplete generalized minimal residual (IGMRES(m)) algorithm based on the fast multipole boundary element method (FM-BEM) was proposed. The convergence theory of the IGMRES(m) algorithm was well established and the related convergence theorem was presented with proof. To further analyze its convergence condition, two numerical experiments were carried out to solve the elastic and elasto-plastic frictional contact problems. Trough the analysis of the influence of truncation index on the computational efficiency and precision, the presented method was proved to be rapidly convergent with stable process and high precision. It was especially suitable for the elastic and elasto-plastic frictional contact problems with complicated and time-consuming iterations.